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“The gambler, the quantum physicist and the juror all reason about probabilities: the probability of winning, of a radioactive atom decaying, of a defendant’s guilt. But despite their ubiquity, experts dispute just what probabilities are. This leads to disagreements on how to reason about, and with, probabilities – disagreements that our cognitive biases can exacerbate, such as our tendency to ignore evidence that runs counter to a hypothesis we favour. Clarifying the nature of probability, then, can help to improve our reasoning.
Three popular theories analyse probabilities as either frequencies, propensities or degrees of belief. Suppose I tell you that a coin has a 50 per cent probability of landing heads up. These theories, respectively, say that this is:
The frequency with which that coin lands heads;
The propensity, or tendency, that the coin’s physical characteristics give it to land heads;
How confident I am that it lands heads.
But each of these interpretations faces problems. Consider the following case:
Adam flips a fair coin that self-destructs after being tossed four times. Adam’s friends Beth, Charles and Dave are present, but blindfolded. After the fourth flip, Beth says: ‘The probability that the coin landed heads the first time is 50 per cent.’
Adam then tells his friends that the coin landed heads three times out of four. Charles says: ‘The probability that the coin landed heads the first time is 75 per cent.’
Dave, despite having the same information as Charles, says: ‘I disagree. The probability that the coin landed heads the first time is 60 per cent.’”
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